Synchronization of Butterﬂy Fractional Order Chaotic System

: In this paper, we study the synchronization of a nonlinear fractional system, and analyze its time response and chaotic behaviors. We represent a solution for considered system by employing the Mittag-Lefﬂer matrix function and Jacobian matrix. Thereafter, we prove synchronization of error system between drive-response systems using stability theory and linear feedback control methods. Finally, numerical simulations are presented to show the effectiveness of the theoretical results.


Introduction
Fractional calculus has over 300 years of history. Recently, it has attracted increasing interest due to its potential applications physics and engineering such as viscoelastic systems [1], dielectric polarization [2], electrode-electrolyte polarization [3], electromagnetic waves [4], quantitative finance [5], and quantum evolution of complex systems [6]. Synchronization of a chaotic system occurs when two or more chaotic systems are connected and it has attracted wide research because of its applications in science and engineering such as biology, economics, secure communication and signal generator design (see [7,8]). Sufficient conditions for synchronization of fractional-order chaotic systems via linear control were investigated in [9], synchronization of fractional chaotic systems with different orders was studied in [10] using stability concepts and synchronization of the Lotka-Volterra chaotic system was studied using the active control method in [11]. Robust synchronization for chaotic and hyperchaotic fractional order systems with model uncertainties and disturbances was investigated in [12].
Fractional chaotic system, global asymptotic synchronization and adaptive sliding mode synchronization have been studied in [13][14][15] and sufficient conditions were presented for exponential synchronization of fractional order chaotic systems in [16]. Finite time stability and synchronization of fractional order chaotic system with uncertainties and disturbance was studied in [17].
In this paper, we consider the following drive system of fractional order and a corresponding response system of (1) is where t ∈ R + := [0, ∞), q ∈ (0, 1), C D q 0 denotes the Caputo fractional derivative [18] with lower limit at 0, u(t) ∈ R n is a state, A is a matrix of dimension n × n, f : R n × R n −→ R n is a C 1 -smooth and h : R + → R n is a controller. Set the error system by e(t) = v(t) − u(t). Define for an n × n matrix B. Then we get is a Jacobian matrix. From [18,19] the solution e(·) ∈ C(R + , R n ) of the system (4) is given by Since E q (−z) and E q,q (−z) are completely monotonous [21], we have for any i = 1, 2, · · · , n and 0 ≤ s ≤ t. So considering any of the following standard norms on R n one can derive the results. (4) is called stable if for any > 0 there exists a δ > 0 such that e 0 < δ guarantees that sup t≥0 e(t) < .

Definition 2.
The drive system (1) is said to be synchronized with the response system (2), if there exists an error system (4) which is stable.

Remark 1.
In [9][10][11][12][14][15][16][17], robust and exponential synchronization of fractional order deterministic chaotic systems was investigated using an adaptive scheme, an active control method, linear control, Lyapunov stability theory, linear feedback controller, sliding mode control and tracking control respectively. In this paper, we investigated the asymptotic synchronization for fractional order system using asymptotic stability theory, a feedback controller and Jacobian matrix.
(ii) The above synchronization is based on the stability of an equilibrium of a linear fractional equation We know that this holds if and only if | arg(λ)| > qπ 2 (12) for any eigenvalue λ of M. But then it is not so clear an estimate like (11). This is a reason, why we consider a diagonal M. For a general M satisfying (12), we must follow the way from Section 3.1 of [23], so a cumbersome approach based on real-valued Jordan form of M. Of course, the order in (7) is not important, so we can consider M = diag (m 11 , m 22 , · · · , m nn ) with m ii < 0 and take λ 1 = max m ii .

Remark 3.
Here, we proposed a key problem to study the synchronization of a nonlinear fractional order system by employing the Mittag-Leffler matrix function, Jacobian matrix, asymptotic stability theory and linear feedback control methods. The main advantages of the consider model is the time response of drive system (1) makes the effect in response system (2) and combined the nonlinear terms by using Jacobian matrix to represent a solution for error system. This type of model is more applicable and reasonable to study the stability concepts via stable equilibrium, point, state, manifold and finite and infinite dimensional stochastic settings. Further, as an applications point of view we show the behaviors of the considered model for different fractional order through numerical simulations.

Examples
In this section, we present the numerical examples to verify the obtained theoretical results.
Example 1. Consider the following fractional order drive system where The state trajectory of the drive system (13) for different fractional orders q = 0.828, 0.9, 0.95, 0.99 are given in Figures 1-4. The trajectories of the system (13) is obviously stable for the fractional order q < 0.828. From  Figures 1-4, one can conclude that the behavior of the state trajectories u 1 (t), u 2 (t), u 3 (t) of the system (13) is unstable for the fractional order q ≥ 0.828.  Thus, it is necessary to introduce the control parameter h(t) to synchronize the error system (15), which is given in the following Example 2.
From Theorem 1, system (13) is synchronized with (14) under the control h(t) as shown in the Figures 12-14. Time responses of the synchronization errors between (13) and (14) are shown in Figure 15, which provides the convergence of the synchronization errors to zero properly. Finally, all the hypothesis of Theorem 1 is verified numerically.  Figure 12. Synchronized time response of the states (t, u 1 (t)) and (t, v 1 (t)) of the drive-response systems (13) and (14) with fractional order q = 0.9.  Figure 13. Synchronized time response of the states (t, u 2 (t)) and (t, v 2 (t)) of the drive-response systems (13) and (14) with fractional order q = 0.9.  Figure 14. Synchronized time response of the states (t, u 3 (t)) and (t, v 3 (t)) of the drive-response systems (13) and (14) with fractional order q = 0.9.  Figure 15. Time response of the states e 1 (t), e 2 (t), e 3 (t) for the error system (15) with fractional order q = 0.9.

Conclusions
We presented synchronization criteria for nonlinear fractional order systems using Jacobian matrix and asymptotic stability estimation of the Mittag-Leffler matrix function, and a suitable linear feedback controller. The above arguments can be extended to fractional evolution equations of (1) and (2) with (3) on a Banach space X when A: D(A) ⊆ X → X is a generator of a C 0 -semigroup {S A (t), t ≥ 0} on X [24], f : X × X → X is globally Lipschitz, h : R + → X is continuous and B : X → X is linear and continuous. We intend to study this case in our next paper.
Author Contributions: M.F. and J.W. contributed to the supervision and project administration, M.F., T.S. and J.W. contributed to the conceptualization and methodology. All authors have read and approved the final manuscript.